16 research outputs found
Golden codes: quantum LDPC codes built from regular tessellations of hyperbolic 4-manifolds
We adapt a construction of Guth and Lubotzky [arXiv:1310.5555] to obtain a
family of quantum LDPC codes with non-vanishing rate and minimum distance
scaling like where is the number of physical qubits. Similarly as
in [arXiv:1310.5555], our homological code family stems from hyperbolic
4-manifolds equipped with tessellations. The main novelty of this work is that
we consider a regular tessellation consisting of hypercubes. We exploit this
strong local structure to design and analyze an efficient decoding algorithm.Comment: 30 pages, 4 figure
Codes correcteurs dâerreur quantique topologiques au-delĂ de la dimension 2
Error correction is the set of techniques used in order to store, process and transmit information reliably in a noisy context. The classical theory of error correction is based on encoding classical information redundantly. A major endeavor of the theory is to find optimal trade-offs between redundancy, which we try to minimize, and noise tolerance, which we try to maximize. The quantum theory of error correction cannot directly imitate the redundant schemes of the classical theory because it has to cope with the no-cloning theorem: quantum information cannot be copied. Quantum error correction is nonetheless possible by spreading the information on more quantum memory elements than would be necessary. In quantum information theory, dilution of the information replaces redundancy since copying is forbidden by the laws of quantum mechanics. Besides this conceptual difference, quantum error correction inherits a lot from its classical counterpart. In this PhD thesis, we are concerned with a class of quantum error correcting codes whose classical counterpart was defined in 1961 by Gallager [Gal62]. At that time, quantum information was not even a research domain yet. This class is the family of low density parity check (LDPC) codes. Informally, a code is said to be LDPC if the constraints imposed to ensure redundancy in the classical setting or dilution in the quantum setting are local. More precisely, this PhD thesis focuses on a subset of the LDPC quantum error correcting codes: the homological quantum error correcting codes. These codes take their name from the mathematical field of homology, whose objects of study are sequences of linear maps such that the kernel of a map contains the image of its left neighbour. Originally introduced to study the topology of geometric shapes, homology theory now encompasses more algebraic branches as well, where the focus is more abstract and combinatorial. The same is true of homological codes: they were introduced in 1997 by Kitaev [Kit03] with a quantum code that has the shape of a torus. They now form a vast family of quantum LDPC codes, some more inspired from geometry than others. Homological quantum codes were designed from spherical, Euclidean and hyperbolic geometries, from 2-dimensional, 3-dimensional and 4- dimensional objects, from objects with increasing and unbounded dimension and from hypergraph or homological products. After introducing some general quantum information concepts in the first chapter of this manuscript, we focus in the two following ones on families of quantum codes based on 4-dimensional hyperbolic objects. We highlight the interplay between their geometric side, given by manifolds, and their combinatorial side, given by abstract polytopes. We use both sides to analyze the corresponding quantum codes. In the fourth and last chapter we analyze a family of quantum codes based on spherical objects of arbitrary dimension. To have more flexibility in the design of quantum codes, we use combinatorial objects that realize this spherical geometry: hypercube complexes. This setting allows us to introduce a new link between classical and quantum error correction where classical codes are used to introduce homology in hypercube complexes.La mĂ©moire quantique est constituĂ©e de matĂ©riaux prĂ©sentant des effets quantiques comme la superposition. Câest cette possibilitĂ© de superposition qui distingue lâĂ©lĂ©ment Ă©lĂ©mentaire de mĂ©moire quantique, le qubit, de son analogue classique, le bit. Contrairement Ă un bit classique, un qubit peut ĂȘtre dans un Ă©tat diffĂ©rent de lâĂ©tat 0 et de lâĂ©tat 1. Une difficultĂ© majeure de la rĂ©alisation physique de mĂ©moire quantique est la nĂ©cessitĂ© dâisoler le systĂšme utilisĂ© de son environnement. En effet lâinteraction dâun systĂšme quantique avec son environnement entraine un phĂ©nomĂšne appelĂ© dĂ©cohĂ©rence qui se traduit par des erreurs sur lâĂ©tat du systĂšme quantique. Dit autrement, Ă cause de la dĂ©cohĂ©rence, il est possible que les qubits ne soient pas dans lâĂ©tat dans lequel il est prĂ©vu quâils soient. Lorsque ces erreurs sâaccumulent le rĂ©sultat dâun calcul quantique a de grandes chances de ne pas ĂȘtre le rĂ©sultat attendu. La correction dâerreur quantique est un ensemble de techniques permettant de protĂ©ger lâinformation quantique de ces erreurs. Elle consiste Ă rĂ©aliser un compromis entre le nombre de qubits et leur qualitĂ©. Plus prĂ©cisĂ©ment un code correcteur dâerreur permet Ă partir de N qubits physiques bruitĂ©s de simuler un nombre plus petit K de qubits logiques, câest-Ă -dire virtuels, moins bruitĂ©s. La famille de codes la plus connue est sans doute celle dĂ©couverte par le physicien Alexei Kitaev: le code torique. Cette construction peut ĂȘtre gĂ©nĂ©ralisĂ©e Ă des formes gĂ©omĂ©triques (variĂ©tĂ©s) autres quâun tore. En 2014, Larry Guth et Alexander Lubotzky proposent une famille de code dĂ©finie Ă partir de variĂ©tĂ©s hyperboliques de dimension 4 et montrent que cette famille fournit un compromis intĂ©ressant entre le nombre K de qubits logiques et le nombre dâerreurs quâelle permet de corriger. Dans cette thĂšse, nous sommes partis de la construction de Guth et Lubotzky et en avons donnĂ© une version plus explicite et plus rĂ©guliĂšre. Pour dĂ©finir un pavage rĂ©gulier de lâespace hyperbolique de dimension 4, nous utilisons le groupe de symĂ©trie de symbole de SchlĂ€fli {4, 3, 3, 5}. Nous en donnons la reprĂ©sentation matricielle correspondant au modĂšle de lâhyperboloĂŻde et Ă un hypercube centrĂ© sur lâorigine et dont les faces sont orthogonales aux quatre axes de coordonnĂ©e. Cette construction permet dâobtenir une famille de codes quantiques encodant un nombre de qubits logiques proportionnel au nombre de qubits physiques et dont la distance minimale croĂźt au moins comme N0.1. Bien que ces paramĂštres soient Ă©galement ceux de la construction de Guth et Lubotzky, la rĂ©gularitĂ© de cette construction permet de construire explicitement des exemples de taille raisonnable et dâenvisager des algorithmes de dĂ©codage qui exploitent cette rĂ©gularitĂ©. Dans un second chapitre nous considĂ©rons une famille de codes quantiques hyperboliques 4D de symbole de SchlĂ€fli {5, 3, 3, 5}. AprĂšs avoir Ă©noncĂ© une façon de prendre le quotient des groupes correspondant en conservant la structure locale du groupe, nous construisons les matrices de paritĂ© correspondant Ă des codes quantiques ayant 144, 720, 9792, 18 000 et 90 000 qubits physiques. Nous appliquons un algorithme de Belief Propagation au dĂ©codage de ces codes et analysons les rĂ©sultats numĂ©riquement. Dans un troisiĂšme et dernier chapitre nous dĂ©finissons une nouvelle famille de codes quantiques Ă partir de cubes de dimension arbitrairement grande. En prenant le quotient dâun cube de dimension n par un code classique de paramĂštres [n, k, d] et en identifiant les qubits physiques avec les faces de dimension p du polytope quotient ainsi dĂ©fini, on obtient un code quantique. Cette famille de codes quantiques a lâoriginalitĂ© de considĂ©rer des quotients par des codes classiques. En cela elle sâĂ©loigne de la topologie et appartient plutĂŽt Ă la famille des codes homologiques
Homological quantum error correcting codes and real projective space
National audienceHomological quantum error correcting codes and real projective spac
Fast erasure decoder for a class of quantum LDPC codes
We propose a decoder for the correction of erasures with hypergraph product
codes, which form one of the most popular families of quantum LDPC codes. Our
numerical simulations show that this decoder provides a close approximation of
the maximum likelihood decoder that can be implemented in O(N^2) bit operations
where N is the length of the quantum code. A probabilistic version of this
decoder can be implemented in O(N^1.5) bit operations.Comment: 5 pages, 2 figure
Single-Shot Decoding of Linear Rate LDPC Quantum Codes with High Performance
We construct and analyze a family of low-density parity check (LDPC) quantum
codes with a linear encoding rate, polynomial scaling distance and efficient
decoding schemes. The code family is based on tessellations of closed,
four-dimensional, hyperbolic manifolds, as first suggested by Guth and
Lubotzky. The main contribution of this work is the construction of suitable
manifolds via finite presentations of Coxeter groups, their linear
representations over Galois fields and topological coverings. We establish a
lower bound on the encoding rate~k/n of~13/72 = 0.180... and we show that the
bound is tight for the examples that we construct. Numerical simulations give
evidence that parallelizable decoding schemes of low computational complexity
suffice to obtain high performance. These decoding schemes can deal with
syndrome noise, so that parity check measurements do not have to be repeated to
decode. Our data is consistent with a threshold of around 4% in the
phenomenological noise model with syndrome noise in the single-shot regime.Comment: 15 pages, 6 figure
Topological Quantum Error-Correcting Codes beyond dimension 2
La mĂ©moire quantique est constituĂ©e de matĂ©riaux prĂ©sentant des effets quantiques comme la superposition. Câest cette possibilitĂ© de superposition qui distingue lâĂ©lĂ©ment Ă©lĂ©mentaire de mĂ©moire quantique, le qubit, de son analogue classique, le bit. Contrairement Ă un bit classique, un qubit peut ĂȘtre dans un Ă©tat diffĂ©rent de lâĂ©tat 0 et de lâĂ©tat 1. Une difficultĂ© majeure de la rĂ©alisation physique de mĂ©moire quantique est la nĂ©cessitĂ© dâisoler le systĂšme utilisĂ© de son environnement. En effet lâinteraction dâun systĂšme quantique avec son environnement entraine un phĂ©nomĂšne appelĂ© dĂ©cohĂ©rence qui se traduit par des erreurs sur lâĂ©tat du systĂšme quantique. Dit autrement, Ă cause de la dĂ©cohĂ©rence, il est possible que les qubits ne soient pas dans lâĂ©tat dans lequel il est prĂ©vu quâils soient. Lorsque ces erreurs sâaccumulent le rĂ©sultat dâun calcul quantique a de grandes chances de ne pas ĂȘtre le rĂ©sultat attendu. La correction dâerreur quantique est un ensemble de techniques permettant de protĂ©ger lâinformation quantique de ces erreurs. Elle consiste Ă rĂ©aliser un compromis entre le nombre de qubits et leur qualitĂ©. Plus prĂ©cisĂ©ment un code correcteur dâerreur permet Ă partir de N qubits physiques bruitĂ©s de simuler un nombre plus petit K de qubits logiques, câest-Ă -dire virtuels, moins bruitĂ©s. La famille de codes la plus connue est sans doute celle dĂ©couverte par le physicien Alexei Kitaev: le code torique. Cette construction peut ĂȘtre gĂ©nĂ©ralisĂ©e Ă des formes gĂ©omĂ©triques (variĂ©tĂ©s) autres quâun tore. En 2014, Larry Guth et Alexander Lubotzky proposent une famille de code dĂ©finie Ă partir de variĂ©tĂ©s hyperboliques de dimension 4 et montrent que cette famille fournit un compromis intĂ©ressant entre le nombre K de qubits logiques et le nombre dâerreurs quâelle permet de corriger. Dans cette thĂšse, nous sommes partis de la construction de Guth et Lubotzky et en avons donnĂ© une version plus explicite et plus rĂ©guliĂšre. Pour dĂ©finir un pavage rĂ©gulier de lâespace hyperbolique de dimension 4, nous utilisons le groupe de symĂ©trie de symbole de SchlĂ€fli {4, 3, 3, 5}. Nous en donnons la reprĂ©sentation matricielle correspondant au modĂšle de lâhyperboloĂŻde et Ă un hypercube centrĂ© sur lâorigine et dont les faces sont orthogonales aux quatre axes de coordonnĂ©e. Cette construction permet dâobtenir une famille de codes quantiques encodant un nombre de qubits logiques proportionnel au nombre de qubits physiques et dont la distance minimale croĂźt au moins comme N0.1. Bien que ces paramĂštres soient Ă©galement ceux de la construction de Guth et Lubotzky, la rĂ©gularitĂ© de cette construction permet de construire explicitement des exemples de taille raisonnable et dâenvisager des algorithmes de dĂ©codage qui exploitent cette rĂ©gularitĂ©. Dans un second chapitre nous considĂ©rons une famille de codes quantiques hyperboliques 4D de symbole de SchlĂ€fli {5, 3, 3, 5}. AprĂšs avoir Ă©noncĂ© une façon de prendre le quotient des groupes correspondant en conservant la structure locale du groupe, nous construisons les matrices de paritĂ© correspondant Ă des codes quantiques ayant 144, 720, 9792, 18 000 et 90 000 qubits physiques. Nous appliquons un algorithme de Belief Propagation au dĂ©codage de ces codes et analysons les rĂ©sultats numĂ©riquement. Dans un troisiĂšme et dernier chapitre nous dĂ©finissons une nouvelle famille de codes quantiques Ă partir de cubes de dimension arbitrairement grande. En prenant le quotient dâun cube de dimension n par un code classique de paramĂštres [n, k, d] et en identifiant les qubits physiques avec les faces de dimension p du polytope quotient ainsi dĂ©fini, on obtient un code quantique. Cette famille de codes quantiques a lâoriginalitĂ© de considĂ©rer des quotients par des codes classiques. En cela elle sâĂ©loigne de la topologie et appartient plutĂŽt Ă la famille des codes homologiques.Error correction is the set of techniques used in order to store, process and transmit information reliably in a noisy context. The classical theory of error correction is based on encoding classical information redundantly. A major endeavor of the theory is to find optimal trade-offs between redundancy, which we try to minimize, and noise tolerance, which we try to maximize. The quantum theory of error correction cannot directly imitate the redundant schemes of the classical theory because it has to cope with the no-cloning theorem: quantum information cannot be copied. Quantum error correction is nonetheless possible by spreading the information on more quantum memory elements than would be necessary. In quantum information theory, dilution of the information replaces redundancy since copying is forbidden by the laws of quantum mechanics. Besides this conceptual difference, quantum error correction inherits a lot from its classical counterpart. In this PhD thesis, we are concerned with a class of quantum error correcting codes whose classical counterpart was defined in 1961 by Gallager [Gal62]. At that time, quantum information was not even a research domain yet. This class is the family of low density parity check (LDPC) codes. Informally, a code is said to be LDPC if the constraints imposed to ensure redundancy in the classical setting or dilution in the quantum setting are local. More precisely, this PhD thesis focuses on a subset of the LDPC quantum error correcting codes: the homological quantum error correcting codes. These codes take their name from the mathematical field of homology, whose objects of study are sequences of linear maps such that the kernel of a map contains the image of its left neighbour. Originally introduced to study the topology of geometric shapes, homology theory now encompasses more algebraic branches as well, where the focus is more abstract and combinatorial. The same is true of homological codes: they were introduced in 1997 by Kitaev [Kit03] with a quantum code that has the shape of a torus. They now form a vast family of quantum LDPC codes, some more inspired from geometry than others. Homological quantum codes were designed from spherical, Euclidean and hyperbolic geometries, from 2-dimensional, 3-dimensional and 4- dimensional objects, from objects with increasing and unbounded dimension and from hypergraph or homological products. After introducing some general quantum information concepts in the first chapter of this manuscript, we focus in the two following ones on families of quantum codes based on 4-dimensional hyperbolic objects. We highlight the interplay between their geometric side, given by manifolds, and their combinatorial side, given by abstract polytopes. We use both sides to analyze the corresponding quantum codes. In the fourth and last chapter we analyze a family of quantum codes based on spherical objects of arbitrary dimension. To have more flexibility in the design of quantum codes, we use combinatorial objects that realize this spherical geometry: hypercube complexes. This setting allows us to introduce a new link between classical and quantum error correction where classical codes are used to introduce homology in hypercube complexes
Codes correcteurs dâerreur quantique topologiques au-delĂ de la dimension 2
Error correction is the set of techniques used in order to store, process and transmit information reliably in a noisy context. The classical theory of error correction is based on encoding classical information redundantly. A major endeavor of the theory is to find optimal trade-offs between redundancy, which we try to minimize, and noise tolerance, which we try to maximize. The quantum theory of error correction cannot directly imitate the redundant schemes of the classical theory because it has to cope with the no-cloning theorem: quantum information cannot be copied. Quantum error correction is nonetheless possible by spreading the information on more quantum memory elements than would be necessary. In quantum information theory, dilution of the information replaces redundancy since copying is forbidden by the laws of quantum mechanics. Besides this conceptual difference, quantum error correction inherits a lot from its classical counterpart. In this PhD thesis, we are concerned with a class of quantum error correcting codes whose classical counterpart was defined in 1961 by Gallager [Gal62]. At that time, quantum information was not even a research domain yet. This class is the family of low density parity check (LDPC) codes. Informally, a code is said to be LDPC if the constraints imposed to ensure redundancy in the classical setting or dilution in the quantum setting are local. More precisely, this PhD thesis focuses on a subset of the LDPC quantum error correcting codes: the homological quantum error correcting codes. These codes take their name from the mathematical field of homology, whose objects of study are sequences of linear maps such that the kernel of a map contains the image of its left neighbour. Originally introduced to study the topology of geometric shapes, homology theory now encompasses more algebraic branches as well, where the focus is more abstract and combinatorial. The same is true of homological codes: they were introduced in 1997 by Kitaev [Kit03] with a quantum code that has the shape of a torus. They now form a vast family of quantum LDPC codes, some more inspired from geometry than others. Homological quantum codes were designed from spherical, Euclidean and hyperbolic geometries, from 2-dimensional, 3-dimensional and 4- dimensional objects, from objects with increasing and unbounded dimension and from hypergraph or homological products. After introducing some general quantum information concepts in the first chapter of this manuscript, we focus in the two following ones on families of quantum codes based on 4-dimensional hyperbolic objects. We highlight the interplay between their geometric side, given by manifolds, and their combinatorial side, given by abstract polytopes. We use both sides to analyze the corresponding quantum codes. In the fourth and last chapter we analyze a family of quantum codes based on spherical objects of arbitrary dimension. To have more flexibility in the design of quantum codes, we use combinatorial objects that realize this spherical geometry: hypercube complexes. This setting allows us to introduce a new link between classical and quantum error correction where classical codes are used to introduce homology in hypercube complexes.La mĂ©moire quantique est constituĂ©e de matĂ©riaux prĂ©sentant des effets quantiques comme la superposition. Câest cette possibilitĂ© de superposition qui distingue lâĂ©lĂ©ment Ă©lĂ©mentaire de mĂ©moire quantique, le qubit, de son analogue classique, le bit. Contrairement Ă un bit classique, un qubit peut ĂȘtre dans un Ă©tat diffĂ©rent de lâĂ©tat 0 et de lâĂ©tat 1. Une difficultĂ© majeure de la rĂ©alisation physique de mĂ©moire quantique est la nĂ©cessitĂ© dâisoler le systĂšme utilisĂ© de son environnement. En effet lâinteraction dâun systĂšme quantique avec son environnement entraine un phĂ©nomĂšne appelĂ© dĂ©cohĂ©rence qui se traduit par des erreurs sur lâĂ©tat du systĂšme quantique. Dit autrement, Ă cause de la dĂ©cohĂ©rence, il est possible que les qubits ne soient pas dans lâĂ©tat dans lequel il est prĂ©vu quâils soient. Lorsque ces erreurs sâaccumulent le rĂ©sultat dâun calcul quantique a de grandes chances de ne pas ĂȘtre le rĂ©sultat attendu. La correction dâerreur quantique est un ensemble de techniques permettant de protĂ©ger lâinformation quantique de ces erreurs. Elle consiste Ă rĂ©aliser un compromis entre le nombre de qubits et leur qualitĂ©. Plus prĂ©cisĂ©ment un code correcteur dâerreur permet Ă partir de N qubits physiques bruitĂ©s de simuler un nombre plus petit K de qubits logiques, câest-Ă -dire virtuels, moins bruitĂ©s. La famille de codes la plus connue est sans doute celle dĂ©couverte par le physicien Alexei Kitaev: le code torique. Cette construction peut ĂȘtre gĂ©nĂ©ralisĂ©e Ă des formes gĂ©omĂ©triques (variĂ©tĂ©s) autres quâun tore. En 2014, Larry Guth et Alexander Lubotzky proposent une famille de code dĂ©finie Ă partir de variĂ©tĂ©s hyperboliques de dimension 4 et montrent que cette famille fournit un compromis intĂ©ressant entre le nombre K de qubits logiques et le nombre dâerreurs quâelle permet de corriger. Dans cette thĂšse, nous sommes partis de la construction de Guth et Lubotzky et en avons donnĂ© une version plus explicite et plus rĂ©guliĂšre. Pour dĂ©finir un pavage rĂ©gulier de lâespace hyperbolique de dimension 4, nous utilisons le groupe de symĂ©trie de symbole de SchlĂ€fli {4, 3, 3, 5}. Nous en donnons la reprĂ©sentation matricielle correspondant au modĂšle de lâhyperboloĂŻde et Ă un hypercube centrĂ© sur lâorigine et dont les faces sont orthogonales aux quatre axes de coordonnĂ©e. Cette construction permet dâobtenir une famille de codes quantiques encodant un nombre de qubits logiques proportionnel au nombre de qubits physiques et dont la distance minimale croĂźt au moins comme N0.1. Bien que ces paramĂštres soient Ă©galement ceux de la construction de Guth et Lubotzky, la rĂ©gularitĂ© de cette construction permet de construire explicitement des exemples de taille raisonnable et dâenvisager des algorithmes de dĂ©codage qui exploitent cette rĂ©gularitĂ©. Dans un second chapitre nous considĂ©rons une famille de codes quantiques hyperboliques 4D de symbole de SchlĂ€fli {5, 3, 3, 5}. AprĂšs avoir Ă©noncĂ© une façon de prendre le quotient des groupes correspondant en conservant la structure locale du groupe, nous construisons les matrices de paritĂ© correspondant Ă des codes quantiques ayant 144, 720, 9792, 18 000 et 90 000 qubits physiques. Nous appliquons un algorithme de Belief Propagation au dĂ©codage de ces codes et analysons les rĂ©sultats numĂ©riquement. Dans un troisiĂšme et dernier chapitre nous dĂ©finissons une nouvelle famille de codes quantiques Ă partir de cubes de dimension arbitrairement grande. En prenant le quotient dâun cube de dimension n par un code classique de paramĂštres [n, k, d] et en identifiant les qubits physiques avec les faces de dimension p du polytope quotient ainsi dĂ©fini, on obtient un code quantique. Cette famille de codes quantiques a lâoriginalitĂ© de considĂ©rer des quotients par des codes classiques. En cela elle sâĂ©loigne de la topologie et appartient plutĂŽt Ă la famille des codes homologiques